Problem: Simplify. Rewrite the expression in the form $a^n$. $a\cdot a^7=$
$\begin{aligned} a\cdot a^7&=a^1\cdot a^7 \\\\ &=a^{1+7} \\\\ &=a^{8} \end{aligned}$ This follows from the general rule $x^m\cdot x^n=x^{m+n}$. Note that the powers have the same base. We can also see this is correct by expanding the powers. $\begin{aligned} a\cdot a^7&=\underbrace{a}_\text{1 time}\cdot\underbrace{a\cdot a\cdot a\cdot a\cdot a\cdot a \cdot a}_\text{7 times} \\\\\\ &=\underbrace{a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a}_\text{8 times} \\\\ &=a^{8} \end{aligned}$ In conclusion, $a\cdot a^7=a^{8}$.